On the polynomial representation of generalized Liouville operators
type: Item
instance of: scholarly article
title: On the polynomial representation of generalized Liouville operators
zbMATH Open document ID: 0773.35002
DOI: 10.1016/S0747-7171(08)80155-1
author: V. Sailer
author: Guillermo Morales Luna
published in: Journal of Symbolic Computation
publication date: 1992-06-28T00:00:00Z
review text: The generalized Liouville operators \({\mathcal L}_ m F=\text{det}\| F_{ij}\|^ m_{i,j=0}\), where \(F_{ij}=\partial^{i+j}F(x,y)/\partial x^ i \partial y^ j\) (\(F(x,y)\) being any infinitely differentiable two-variable function) is considered. The problem of finding a polynomial representation for \({\mathcal L_ n}({\mathcal L}_ m F)\) by the means of algebraic manipulations is analysed. Using computer programs the authors expand the Liouville expressions into sums of individual terms via direct computation. The algorithms recognize groups of isolated terms which cannot be represented completely as polynomials over the Liouville operators. It is shown that no general polynomial expression can exist for the composite operation \({\mathcal L}_ n({\mathcal L}_ m F)\).
Mathematics Subject Classification ID: 35-04
Mathematics Subject Classification ID: 35G20
zbMATH DE Number: 37007
zbMATH Keywords: computer algorithms
zbMATH Keywords: operator composition
zbMATH Keywords: generalized Liouville operators
reviewed by: Vladimir L. Makarov
MaRDI profile type: Publication
cites work: Wronskians, geometry, and some general solutions to the nonlinear Liouville-like PDE’s
description: scientific article; zbMATH DE number 37007